If `A=[{:(,3,x),(,0,1):}] and B=[{:(,9,16),(,0,-y):}]` find x and y when `A^2=B`.

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that \( A^2 = B \), where:

\[ A = \begin{pmatrix} 3 & x \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 9 & 16 \\ 0 & -y \end{pmatrix} \]

### Step 1: Calculate \( A^2 \)

We start by calculating \( A^2 \) which is \( A \times A \):

\[ A^2 = \begin{pmatrix} 3 & x \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 3 & x \\ 0 & 1 \end{pmatrix} \]

### Step 2: Perform the Matrix Multiplication

Using the rules of matrix multiplication, we calculate each element of the resulting matrix:

- First row, first column:
\[ 3 \cdot 3 + x \cdot 0 = 9 \]

- First row, second column:
\[ 3 \cdot x + x \cdot 1 = 3x + x = 4x \]

- Second row, first column:
\[ 0 \cdot 3 + 1 \cdot 0 = 0 \]

- Second row, second column:
\[ 0 \cdot x + 1 \cdot 1 = 1 \]

Putting it all together, we get:

\[ A^2 = \begin{pmatrix} 9 & 4x \\ 0 & 1 \end{pmatrix} \]

### Step 3: Set \( A^2 \) Equal to \( B \)

According to the problem, we have:

\[ A^2 = B \]

So we set:

\[ \begin{pmatrix} 9 & 4x \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 9 & 16 \\ 0 & -y \end{pmatrix} \]

### Step 4: Equate Corresponding Elements

Now we equate the corresponding elements from both matrices:

1. From the first row, first column:
\[ 9 = 9 \quad \text{(This is always true)} \]

2. From the first row, second column:
\[ 4x = 16 \]

3. From the second row, second column:
\[ 1 = -y \]

### Step 5: Solve for \( x \) and \( y \)

Now we solve the equations:

1. From \( 4x = 16 \):
\[ x = \frac{16}{4} = 4 \]

2. From \( 1 = -y \):
\[ y = -1 \]

### Final Result

Thus, the values of \( x \) and \( y \) are:

\[ x = 4, \quad y = -1 \]